Aligning the community college math curriculum with the Common

Aligning the community
college math curriculum
with the Common Core
State Standards in Math
Bruce Yoshiwara
Los Angeles Pierce College
Math Curriculum Framework and Evaluation Criteria Committee
What are the Common
Core State Standards?
“Educational standards [that] describe what
students should know and be able to do in each
subject in each grade. In California, the State
Board of Education decides on the standards for
all students, from kindergarten through high
school.”
http://www.cde.ca.gov/re/cc/tl/whatareccss.asp
What are the Common
Core State Standards?
(continued)
Forty-five states, the District of Columbia, four
territories, and the Department of Defense
Education Activity have adopted the Common
Core State Standards.
http://www.corestandards.org/Math
What are the Common
Core State Standards?
(continued)
California is part of the Smarter Balanced
Assessment Consortium (SBAC). SBAC uses
computer-based, adaptive testing.
The other major CCSS assessment consortium
is the Partnership for Assessment of Readiness
for College and Careers (PARCC)
How are CA community
colleges affected by the
CCSS in Math?
Not only will our incoming students be
differently prepared, but
How are CA community
colleges affected by the
CCSS in Math?
Not only will our incoming students be
differently prepared, but our articulation
and transfer agreements with four-year
schools may be altered by the changed
expectations of what it means to be
“college and career ready.”
BOARS clarified (December 2013)
that “… going forward, all students
must complete the basic
mathematics defined by the
college-ready standards of the
Common Core State Standards for
Mathematics (CCSSM) prior to
enrolling in a UC-transferable
college mathematics or statistics
course.” [Emphasis mine]
How are CA community
colleges affected by the
CCSS in Math?
(continued)
Our assessment and placement instruments will
need adjustment or replacement.
How are CA community
colleges affected by the
CCSS in Math?
(continued)
Our assessment and placement instruments will
need adjustment or replacement. (The existing
CA Early Assessment Program exempts
students who meet a set score on the 11th
grade assessment from taking placement
exams in CCCs and certifies that these students
are ready for transfer-level math courses.)
So what’s in the CCSSM?
 Standards
for
Mathematical Practice
 Standards for
Mathematical Content
8 Standards for
Mathematical Practice
The MP describe how math students
ought to engage with the subject matter
as they grow in mathematical maturity
and expertise throughout the
elementary, middle ,and high school
years.
8 Standards for
Mathematical Practice
1) Make sense of problems
and persevere in solving
them.
8 Standards for
Mathematical Practice
2) Reason abstractly and
quantitatively.
8 Standards for
Mathematical Practice
3) Construct viable
arguments and critique the
reasoning of others.
8 Standards for
Mathematical Practice
4) Model with mathematics.
8 Standards for
Mathematical Practice
5) Use appropriate tools
strategically.
8 Standards for
Mathematical Practice
6) Attend to precision.
8 Standards for
Mathematical Practice
7) Look for and make use of
structure.
8 Standards for
Mathematical Practice
8) Look for and express
regularity in repeated
reasoning.
Standards for
Mathematical Content
There are content standards at each
K-8 grade level.
The “Higher Mathematics” (a.k.a.
high school) content standards are
grouped into 6 categories.
6 Conceptual
Categories for higher
mathematics
1.
2.
3.
4.
5.
6.
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
“The higher mathematics standards specify
the mathematics that all students should
study in order to be college and career
ready.
“The higher mathematics standards specify
the mathematics that all students should
study in order to be college and career
ready. Additional mathematics that students
should learn in preparation for advanced
courses, such as calculus, advanced
statistics, or discrete mathematics, is
indicated by a plus symbol (+). All
standards without a (+) symbol should be
in the common mathematics curriculum for
all college and career ready students.
Standards with a (+) symbol may also
appear in courses intended for all students.”
Higher Mathematics
Content Standards
The CA CCSSM suggests two possible
pathways to include the higher
mathematics content standards: a
Traditional Pathway (Algebra I, Geometry,
and Algebra II) and an Integrated
Pathway (Mathematics I, II, and III).
CCSS Algebra I
includes some nontraditional topics
Linear, quadratic, and exponential
functions, including arithmetic and
geometric sequences as functions,
function notation, and fitting
functions to data
CCSS Algebra I
includes some nontraditional topics
Statistics, including assessing the fit
of a function by plotting and analyzing
residuals; interpreting the slope,
intercept, and correlation coefficient
of a linear model in context.
CCSS Geometry
includes some nontraditional topics
Transformational geometry:
congruence defined in terms of
rigid motion; similarity defined in
terms of dilations and rigid
motions.
CCSS Geometry
includes some nontraditional topics
Trigonometry: trig ratios, special
angles, derivation of the equation
of a parabola given a focus and
directrix.
CCSS Geometry
includes some nontraditional topics
Probability: sample spaces,
independent events, conditional
probability, permutations and
combinations; analyzing decisions
and strategies using probability
CCSS Algebra II
includes some nontraditional topics
Trigonometry: 6 trig functions of
real numbers; modeling periodic
phenomena, proof and use of the
identity sin 2   cos2   1
CCSS Algebra II
includes some nontraditional topics
Statistics: normal distributions,
random samples, estimating
population parameters,
simulations, using probability to
make decisions
Examples of (+)
standards in traditional
intermediate Algebra
(+) Find the conjugate of a complex number; use conjugates
to find moduli and quotients of complex numbers.
(+) Extend polynomial identities to the complex numbers. For
example, rewrite x2 + 4 as (x + 2i)(x – 2i).
(+) Know the Fundamental Theorem of Algebra; show that it
is true for quadratic polynomials.
Examples of (+)
standards in traditional
intermediate Algebra
(+) Know and apply the Binomial Theorem for the
expansion of (x + y)n in powers of x and y for a
positive integer n, where x and y are any numbers,
with coefficients determined for example by
Pascal’s Triangle
Examples of (+)
standards in traditional
intermediate Algebra
(+) Understand that rational expressions form a
system analogous to the rational numbers, closed
under addition, subtraction, multiplication, and
division by a nonzero rational expression; add,
subtract, multiply, and divide rational expressions.
Examples of (+)
standards in traditional
intermediate Algebra
(+) Graph rational functions, identifying zeros and
asymptotes when suitable factorizations are
available, and showing end behavior.
Examples of (+)
standards in traditional
intermediate Algebra
(+) Verify by composition that one function is the
inverse of another.
(+) Read values of an inverse function from a
graph or a table, given that the function has an
inverse.
Examples of (+)
standards in traditional
intermediate Algebra
(+) Produce an invertible function from a noninvertible function by restricting the domain.
(+) Understand the inverse relationship between
exponents and logarithms and use this relationship
to solve problems involving logarithms and
exponents.
More (+) standards
Geometry of complex numbers (3 standards)
Vectors (5 standards)
Matrices (9 standards)
Trig (6 standards)
Geometry (3 standards)
Probability/stats (9 standards)
Do CCCs need to align
with CCSSM?
The Student Success Task
Force recommends it:
"Aligning K-12 and community colleges
standards for college and career readiness
is a long-term goal that will require a
significant investment of time and energy
that the Task Force believes will pay off by
streamlining student transition to college
and reducing the academic deficiencies of
entering students…
"Aligning K-12 and community colleges
standards for college and career readiness
is a long-term goal that will require a
significant investment of time and energy
that the Task Force believes will pay off by
streamlining student transition to college
and reducing the academic deficiencies of
entering students…
"Recommendation 1.1: Community
Colleges will collaborate with K-12
education to jointly develop new common
standards for college and career readiness
that are aligned with high school exit
standards."
Do CCCs need to align
with CCSSM?
(continued)
The University of California
expects it:
The UC Board of Admissions and
Relations with Schools (BOARS)
wrote in July 2013 that “… the
basic mathematics of the CCSSM
can appropriately be used to
define the minimal level of
mathematical competence that
all incoming UC students should
demonstrate.”
BOARS clarified (December 2013)
that “… going forward, all students
must complete the basic
mathematics defined by the
college-ready standards of the
Common Core State Standards for
Mathematics (CCSSM) prior to
enrolling in a UC-transferable
college mathematics or statistics
course.”
What does the UC mean
by CCSSM “alignment”?
“Much of the longstanding discussion surrounding
what foundational mathematics is necessary for
college-level mathematics focuses on algebra. But
it is important to note that algebra is only one of
several topics identified in the CCSSM. Also
specified are number and quantity, functions,
modeling, geometry, and statistics and
probability.” (12/2013)
What does the UC mean
by CCSSM “alignment”?
(continued)
“Specifying that transferable courses must
have at least Intermediate Algebra as a
prerequisite is not fully consistent with the
use of the basic mathematics of the CCSSM
as a measure of college readiness ….”
(7/13)
What does the UC mean
by CCSSM “alignment”?
(continued)
“Specifying that transferable courses must
have at least Intermediate Algebra as a
prerequisite is not fully consistent with the
use of the basic mathematics of the CCSSM
as a measure of college readiness in that
most existing Intermediate Algebra courses
contain topics that are identified in the
CCSSM as part of the (+) standards.” (7/13)
What does the UC mean
by CCSSM “alignment”?
(continued)
“Because current course offerings of Intermediate Algebra include
material identified in the CCSSM as “additional mathematics that
students should learn in order to take advanced courses such as
calculus, advanced statistics, or discrete mathematics,” it will not be
appropriate in the future to use traditional Intermediate Algebra
(i.e., Intermediate Algebras as defined prior to CCSSM
implementation) as the primary standard for demonstrating the
minimal level of mathematical competence that BOARS seeks in
students admitted to UC.” (7/13)
What does the UC mean
by CCSSM “alignment”?
(continued)
“Requiring that all prospective transfer students pass the
current version of Intermediate Algebra would be asking
more of them than UC will ask of students entering as
freshmen who have completed CCSSM-aligned high school
math courses…”
(7/13)
What does the UC mean
by CCSSM “alignment”?
(continued)
“Requiring that all prospective transfer students pass the
current version of Intermediate Algebra would be asking
more of them than UC will ask of students entering as
freshmen who have completed CCSSM-aligned high school
math courses. As such, BOARS expects that the Transferable
Course Agreement Guidelines will be rewritten to clarify that
the prerequisite mathematics for transferable courses should
align with the college-ready content standards of the
CCSSM.” (7/13)
What does the UC mean
by CCSSM “alignment”?
(continued)
“BOARS acknowledges that the continued use of
Intermediate Algebra as the prerequisite for UC-transferable
courses is problematic. Such courses traditionally cover more
advanced topics than are included in the basic college-ready
CCSSM standards. ...”
(12/2013)
What does the UC mean
by CCSSM “alignment”?
(continued)
“BOARS acknowledges that the continued use of
Intermediate Algebra as the prerequisite for UC-transferable
courses is problematic. Such courses traditionally cover more
advanced topics than are included in the basic college-ready
CCSSM standards. Thus, BOARS’s statement closes with the
expectation that future UC-transferable courses will have
prerequisites that align with the Common Core, not
prerequisites that have a particular name.” (12/2013)
What does the UC mean
by CCSSM “alignment”?
(continued)
“BOARS recognizes that this is a period of transition in
mathematics instruction, moving from traditional course
sequences to new courses and sequences. Within the CCSSM,
there are multiple pathways to meet the college-ready
standards, and BOARS encourages the development of such
new approaches within the California Community Colleges…”
(12/2013)
What does the UC mean
by CCSSM “alignment”?
(continued)
“BOARS recognizes that this is a period of transition in
mathematics instruction, moving from traditional course
sequences to new courses and sequences. Within the CCSSM,
there are multiple pathways to meet the college-ready
standards, and BOARS encourages the development of such
new approaches within the California Community Colleges.
The key is to ensure that students have met the standards of
the Common Core State Standards for Mathematics, not that
they have completed a specific course.” (12/2013)
What does the UC mean
by CCSSM “alignment”?
(continued)
According to the July UC BOARS statement:
“The most recent version of the ICAS
mathematical competency statement makes
clear the close alignment between it and the
CCSSM. Both define the mathematics that all
students should study in order to be college
ready.” [Emphasis mine]
The Intersegmental Committee of the Academic
Senates: “The goal of this Statement on
Competencies in Mathematics Expected of Entering
College Students is to provide a clear and coherent
message about the mathematics that students need
to know and to be able to do to be successful in
college. ”
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Right triangle trigonometry;
transformational geometry,
including dilations. (ICAS
lists only as “desirable”)
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Solutions to systems of equations and their
geometrical interpretation; solutions to quadratic
equations, both algebraic and graphical; complex
numbers and their arithmetic; the correspondence
between roots and factors of polynomials; rational
expressions; the binomial theorem. (ICAS lists
only for STEM)
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Trigonometric functions of real variables,
their graphs, properties including
periodicity, and applications to right
triangle trigonometry; basic trigonometric
identities. (ICAS lists only for STEM)
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Two- and three-dimensional coordinate
geometry; locus problems. (ICAS lists only
for STEM)
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Distributions as models; the Normal
Distribution; fitting data with curves;
correlation, regression; sampling,
graphical displays of data. (ICAS lists
only for STEM)
What ICAS does NOT
consider necessary for
all students (but is in
CCSSM):
Conic sections: representations as
plane sections of a cone; focusdirectrix properties; reflective
properties. (ICAS lists only for STEM)
Aside: The National Center on Education
and the Economy (May 2013)
“Mastery of Algebra II is widely
thought to be a prerequisite for
success in college and careers.
Our research shows that that is
not so... Based on our data, one
cannot make the case that high
school graduates must be
proficient in Algebra II to be
ready for college and careers.”
http://www.ncee.org/college-and-work-ready/
What does the UC mean
by CCSSM “alignment”?
(continued)
The ICAS mathematical
competency statement begins
with “Part 1: Dispositions of
well-prepared students toward
mathematics.”
•A view that mathematics
makes sense—students should
perceive mathematics as a way
of understanding, not as a
sequence of algorithms to be
memorized and applied.
•An ease in using their mathematical
knowledge to solve unfamiliar problems in
both concrete and abstract situations—
students should be able to find patterns,
make conjectures, and test those
conjectures; they should recognize that
abstraction and generalization are important
sources of the power of mathematics; they
should understand that mathematical
structures are useful as representations of
phenomena in the physical world; they
should consistently verify that their solutions
to problems are reasonable.
•A willingness to work on mathematical
problems requiring time and thought,
problems that are not solved by merely
mimicking examples that have already
been seen—students should have
enough genuine success in solving
such problems to be confident, and
thus to be tenacious, in their approach
to new ones.
•A readiness to discuss the
mathematical ideas involved in a
problem with other students and to
write clearly and coherently about
mathematical topics—students should
be able to communicate their
understanding of mathematics with
peers and teachers using both formal
and natural languages correctly and
effectively.
•An acceptance of responsibility for
their own learning—students
should realize that their minds are
their most important mathematical
resource, and that teachers and
other students can help them to
learn but can’t learn for them.
•The understanding that assertions
require justification based on
persuasive arguments, and an ability
to supply appropriate justifications—
students should habitually ask “Why?”
and should have a familiarity with
reasoning at a variety of levels of
formality, ranging from concrete
examples through informal arguments
using words and pictures to precise
structured presentations of convincing
arguments.
•While proficiency in the use of
technology is not a substitute for
mathematical competency,
students should be familiar with
and confident in the use of
computational devices and
software to manage and display
data, to explore functions, and to
formulate and investigate
mathematical conjectures.
•A perception of mathematics as a
unified field of study—students
should see interconnections among
various areas of mathematics,
which are often perceived as
distinct.
CCSSM Mathematical Practices
1. Make sense of problems and persevere
in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and
critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in
repeated reasoning.
What does the UC mean
by CCSSM “alignment”?
(continued)
The ICAS math Dispositions and
the CCSSM standards for
Mathematical Practice are
consistent…is that sufficient for
“close alignment”?
The entire CCSSM
document and
ancillaries are
available for free
download from the
CA Dept of Ed
website:
http://www.cde.ca.gov/re/cc/
The approved draft
of the CA Math
Framework is also
available online:
http://www.cde.ca.gov/ci/ma/cf/
Thank you!
Bruce Yoshiwara
(A Google search on my name
should find my homepage, and from
there links to handouts/information
for faculty, including this file and
links to related resources.)
“The eight Standards for Mathematical
Practice (MP) describe the attributes of
mathematically proficient students and
expertise that mathematics educators at
all levels should seek to develop in their
students. Mathematical practices provide a
vehicle through which students engage
with and learn mathematics. As students
move from elementary school through
high school, mathematical practices are
integrated in the tasks as students engage
in doing mathematics and master new and
more advanced mathematical ideas and
understandings.” (CA Framework)
More (+) standards
(+) Represent complex numbers on the complex
plane in rectangular and polar form (including real
and imaginary numbers), and explain why the
rectangular and polar forms of a given complex
number represent the same number.
More (+) standards
(+) Represent addition, subtraction, multiplication, and
conjugation of complex numbers geometrically on the
complex plane; use properties of this representation for
computation. For example, (–1 + √3 i)3 = 8 because (–1 +
√3 i) has modulus 2 and argument 120°.
(+) Calculate the distance between numbers in the complex
plane as the modulus of the difference, and the midpoint of a
segment as the average of the numbers at its endpoints.
More (+) standards
Represent and model with vector quantities
(3 standards).
Perform operations on vectors (2 standards).
More (+) standards
(+) Use matrices to represent and manipulate
data, e.g., to represent payoffs or incidence
relationships in a network.
(+) Multiply matrices by scalars to produce new
matrices, e.g., as when all of the payoffs in a game
are doubled.
More (+) standards
(+) Add, subtract, and multiply matrices of
appropriate dimensions.
(+) Understand that, unlike multiplication of
numbers, matrix multiplication for square matrices
is not a commutative operation, but still satisfies
the associative and distributive properties.
More (+) standards
(+) Understand that the zero and identity matrices play a role in matrix
addition and multiplication similar to the role of 0 and 1 in the real
numbers. The determinant of a square matrix is nonzero if and only if
the matrix has a multiplicative inverse.
(+) Multiply a vector (regarded as a matrix with one column) by a
matrix of suitable dimensions to produce another vector. Work with
matrices as transformations of vectors.
(+) Work with 2 × 2 matrices as transformations of the plane, and
interpret the absolute value of the determinant in terms of area.
More (+) standards
(+) Represent a system of linear equations as a
single matrix equation in a vector variable.
(+) Find the inverse of a matrix if it exists and use
it to solve systems of linear equations (using
technology for matrices of dimension 3 × 3 or
greater).
More (+) standards
(+) Understand that restricting a trigonometric function
to a domain on which it is always increasing or always
decreasing allows its inverse to be constructed.
(+) Use inverse functions to solve trigonometric
equations that arise in modeling contexts; evaluate the
solutions using technology, and interpret them in terms
of the context.
More (+) standards
(+) Prove the addition and subtraction formulas for
sine, cosine, and tangent and use them to solve
problems.
(+) Derive the formula A = 1/2 ab sin(C) for the
area of a triangle by drawing an auxiliary line from
a vertex perpendicular to the opposite side.
More (+) standards
(+) Prove the Laws of Sines and Cosines and use
them to solve problems.
(+) Understand and apply the Law of Sines and the
Law of Cosines to find unknown measurements in
right and non-right triangles (e.g., surveying
problems, resultant forces).
More (+) standards
(+) Construct a tangent line from a point outside a given
circle to the circle.
(+) Derive the equations of ellipses and hyperbolas given the
foci, using the fact that the sum or difference of distances
from the foci is constant.
(+) Give an informal argument using Cavalieri’s principle for
the formulas for the volume of a sphere and other solid
figures.
More (+) standards
(+) Apply the general Multiplication Rule in a uniform
probability model, P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the answer in terms of the
model.
(+) Use permutations and combinations to compute
probabilities of compound events and solve problems.
More (+) standards
Calculate expected values and use them to
solve problems (4 standards)
Use probability to evaluate outcomes of
decisions (3 standards)